Calculus BC AP

Limts

Notation for: Limit from the left of $f(x)$ as $x \to a$ $$\lim _{x \to a^-} {f(x)}$$

Limit from the right of $f(x)$ as $x \to a$ $$\lim _{x \to a^+} {f(x)}$$

$\lim _{x\to a} {f(x)} = F$ and $\lim _{x \to a} {g(x)} = G$

$$\lim _{x \to a} {g(f(x))} = g(\lim _{x \to a} f(x)) = g(F)$$

$$\lim _{x \to a} \frac {f(x)} {g(x)} = \frac {\lim _{x \to a} f(x)} {\lim _{x \to a} g(x)}$$

Special Cases Break anywhere, split into right & left hand?

Def. Of Cont.

$lim _{x\to a} f(x)$ exists, $f(a)$, $lim _{x\to a} f(x) = f(a)$

Conditions in which a limit DNE

1. Sharp Turn
2. Cusp
3. Vertical Tangent
4. Discont.

Derivatives

Average Rate Of Change (= Slope) $$\frac{f(B) - f(A)}{B-A}$$

Limit Def. of Derivative $$\frac{d}{dx} (f(x)) = \lim _{h \to 0} \frac {f(x + h) - f(x)} h$$

Alternative Limit Def. of Deriv. $${d \over dx} (f(x)) = \lim _{x \to a} \frac {f(x) - f(a)} {x - a}$$

Chain Rule $${d \over dx} [f(g(x))] = f'(g(x)) * g'(x) = (f' \circ g) (x) * g'(x)$$

Product Rule $${d \over dx} (f(x)*g(x)) = f(x)g'(x) + f'(x)g(x)$$

Quotient Rule $${d \over dx} \left[{h(x) \over k(x)}\right] = {k(x)h'(x)-h(x)k'(x) \over (k(x))^2} = {\text {Lo dHi} - \text {Hi dLo} \over \text {Lo}^2}$$

Derivatives of Inverse Functions

$$f(x) \Rightarrow \frac 1 {f'(y)}$$

Linear Motion

Position: $s(t)$ Velocity: $v(t) = s'(t)$ Acceleration: $a(t) = v'(t) = s''(t)$

Derivatives++

$\frac{dx}{dy} \left(x^n\right) = nx^{n-1}$ $\frac{dx}{dy} \left(\sin x\right) = \cos (x)$ $\frac{dx}{dy} \left(\cos x\right) = -\sin (x)$ $\frac{dx}{dy} \left(\tan x\right) = \sec ^2 (x)$ $\frac{dx}{dy} \left(\cot x\right) = -\csc ^2 (x)$ $\frac{dx}{dy} \left(\sec x\right) = \sec (x)\tan (x)$ $\frac{dx}{dy} \left(\csc x\right) = -\csc (x)\cot (x)$ $\frac{dx}{dy} \left(\ln x\right) = \frac{1}{x}$ $\frac{dx}{dy} \left(e^{x}\right) = e^x$ $\frac{dx}{dy} \left(\sin ^{-1} x\right) = {1 \over \sqrt {1 - x ^2}}$ $\frac{dx}{dy} \left(\sin ^ {-1} x\right) = {1 \over 1 + x^2}$ $\frac{dx}{dy} \left(\sec ^{-1} x\right) = {1 \over |x| \sqrt {x^{2}- 1}}$ $\frac{dx}{dy} \left(\cos ^{-1} x\right) = {-1 \over \sqrt {1 - x^2}}$ $\frac{dx}{dy} \left(a^x\right) = \ln(a)a^x$ $\frac{dx}{dy} \left(\log _a x\right) = {1 \over \ln(a) x}$

Intermediate Value Theorem

If cont. on $[A, B]$ then every value between $A$ and $B$ exist.

Mean Value Theorem

If cont. on $[A, B]$ and differentiable on $(A, B)$ then $f'(c) = {f(B) - f(A) \over B - A}$ for some value of $c$

First Fundamental Theorem of Calculus

$$\int _A ^B f'(x) dx = f(B) - f(A)$$

Second Fundamental Theorem of Calculus

$$\frac {d} {dx} \left[\int _A ^{g(x)} f(t) dt\right] = f(g(x))g'(x)$$

Average Value

$$f_{avg} = {1 \over B - A} \int _A ^B f(x) dx$$

Solids of Revolution & Co.

Cross Sections

$$\int A(x)dx$$ $A(x) =$ Area of shape

Disk Method

$$\pi \int (r(x))^2dx$$ $r(x) =$ radius

Washer Method

$$\pi \int \left(R(x)\right)^2 - (r(x))^2 dx$$

$R(x) =$ Outer radius $r(x) =$ Inner radius

Arc Length (Rectangular)

==COMPLETE ME==

L'Hopital's Rule

If

$$\lim _{x \to a} {f(x) \over g(x)} \Rightarrow {0 \over 0} \text{ or } {\infty \over \infty}$$

then

$$\lim _{x \to a} {f(x) \over g(x)} = L'Hop \lim _{x \to a} {f'(x) \over g'(x)}$$

Advanced Integration Techniques

1. U-Sub
2. Logn Division
3. Complete the Square
4. Partial Fractions
5. [[#Integration by Parts]]

Integration by Parts

$$\int u \, dv = uv - \int v \, du$$

Logistic Growth

$$\begin{align*} { dP \over dt } = KP(L-P)\\ P = { L \over 1 + Q e^{-klt}} \end{align*}$$

Parametric Equations

$$\begin{align*} { dy \over dx } = { dy / dt \over dx / dt }\\ { d^2y \over dx^2 } = { \frac {d} {dt} \left[{ dy \over dx }\right] \over dx/dt } \end{align*}$$

Taylor Series

$$\sum\limits _{n=0} ^{\infty} { f^{(n)}(c) \over n! }(x-c)^n$$

Maclaurin Series

$$ \begin{align} e^{ x } &= \sum _{n=0} ^\infty \frac{x^n}{n!} &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \

\cos x &= \sum _{n=0} ^\infty \frac{(-1)^{n}x^{2n}}{(2n)!} &= 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\dots \

\sin x &= \sum _{n=0} ^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!} &= x - \frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\dots \

\frac{1}{1-x} &= \sum {n=0} ^\infty x^n &= \underbrace{ 1 + x + x^{2} + x^{3} + \dots }{ \text{Expanded Form} }

\end{align}

$$